Deck 5: Classical Linear Regression Model Assumptions and Diagnostic Tests

Question 1

Assuming that the restricted sum of squares of the restricted regression in question 3 is 436.1 and the unrestricted sum of squares is 397.2, what would the conclusion of the hypothesis test be? (The significance level is 5%)

A) Reject the null hypothesis
B) Do not reject the null hypothesis
C) Reject the alternative hypothesis
D) Cannot say

Answer 1

Reject the null hypothesis

Question 2

Question refers to the following regression estimated on 64 observations:
Y_t = $\beta$ ₁ + $\beta$ ₂X_2t + $\beta$ ₃X_3t + $\beta$ ₄X_4t + u_t

-Which of the following null hypotheses could we test using an F-test?
(I) $\beta$ ₂ = 0
(ii) $\beta$ ₂ = 1 and $\beta$ _{3 +} $\beta$ ₄ = 1
(iii) $\beta$ ₃ $\beta$ ₄ = 1
(iv) $\beta$ ₂ - $\beta$ ₃ - $\beta$ ₄ = 1

A) (i) and (ii) only
B) (ii) and (iv) only
C) (i), (ii), (iii) and (iv)
D) (i), (ii), and (iv) only

Answer 2

(i), (ii), and (iv) only

Question 3

Which one of the following is the most appropriate as a definition of R² in the context that the term is usually used?

A) It is the proportion of the total variability of y that is explained by the model
B) It is the proportion of the total variability of y about its mean value that is explained by the model
C) It is the correlation between the fitted values and the residuals
D) It is the correlation between the fitted values and the mean.

Answer 3

It is the proportion of the total variability of y about its mean value that is explained by the model

Question 4

Consider the following two regressions Which of the following statements are true?
(I) The RSS will be the same for the two models
(ii) The R² will be the same for the two models
(iii) The adjusted R² will be different for the two models
(iv) The regression F-test will be the same for the two models

A) (ii) and (iv) only
B) (i) and (iii) only
C) (i), (ii), and (iii) only
D) (i), (ii), (iii), and (iv).

Question 5

Why is R² a commonly used and perhaps better measure of how well a regression model fit the data than the residual sum of squares (RSS)?

A) The RSS is often too large
B) The RSS does not depend on the scale of the dependent variable whereas the R² does
C) The RSS depends on the scale of the dependent variable whereas the R² does not
D) The RSS depends on the scale of the independent variable whereas the R² does not

Question 6

Suppose that the value of R² for an estimated regression model is exactly one. Which of the following are true?
(I) All of the data points must lie exactly on the line
(ii) All of the residuals must be zero
(iii) All of the variability of y about is mean have has been explained by the model
(I) The fitted line will be horizontal with respect to all of the explanatory variables

A) (ii) and (iv) only
B) (i) and (iii) only
C) (i), (ii), and (iii) only
(I), (ii), (iii), and (iv)

Question 7

What would the restricted regression be if you are interested in testing the null hypothesis and against the alternative hypothesis or for a regression ,?

A)
B)
C)
D)

Question 8

Use the following to answer questions
Assuming you have two regression models and .
How can the two models be validly compared to determine the model that better represents the data y_t?

A) By observing their respective R²
B) By observing their respective Adjusted R²
C) By estimating an encompassing or hybrid model
D) All of the above

Question 9

Consider the following regression estimated using 84 observations: y_t = $\beta$ ₁ + $\beta$ ₂X_2t + $\beta$ ₃X_3t + $\beta$ ₄X_4t + u_t
Suppose that a researcher wishes to test the null hypothesis: $\beta$ ₂ = 1 and $\beta$ _{3 +} $\beta$ ₄ = 1. The TABULATED value of the F-distribution that we would compare the result of testing this hypothesis with at the 10% level would be approximately

A) 19.48
B) 2.76
C) 2.37
D) 3.11

Question 10

Which of the following are often considered disadvantages of the use of adjusted R² as a variable addition / variable deletion rule?
(I) Adjusted R² always rises as more variables are added
(ii) Adjusted R² often leads to large models with many marginally significant or marginally insignificant variables
(iii) Adjusted R² cannot be compared for models with different explanatory variables
(iv) Adjusted R² cannot be compared for models with different explained variables.

A) (ii) and (iv) only
B) (i) and (iii) only
C) (i), (ii), and (iii) only
D) (i), (ii), (iii), and (iv).

Question 11

Which one of the following statements must hold for EVERY CASE concerning the residual sums of squares for the restricted and unrestricted regressions?

A) URSS > RRSS
B) URSS $\ge$ RRSS
C) RRSS > URSS
D) RRSS $\ge$ URSS

Question 12

If our regression equation is y = X $\beta$ + u, where we have T observations and k regressors, what will be the dimension of using the standard matrix notation

A) T * k
B) T * 1
C) k *1
D) k * k

Question 13

Use the following to answer questions
Assuming you have two regression models and .
What is the relevant encompassing model required to compare the two regression models?

A)
B)
C)
D) Encompassing models cannot be used to compare these specifications

Question 14

Which of these statements is a characteristic of the stepwise regression procedure?
(I) It chooses the jointly most 'important' explanatory variable from a set of candidate variables
(II) It can start with no variables in the regression and then it selects first the variable with the lowest p-value
(III) It can start with no variables in the regression and then it selects first the variable with the highest p-value

A) I only
B) II only
C) III only
D) Both I and II

Question 15

Which of these statements is not true about quantile regressions?

A) No distributional assumptions are required to optimally estimate the parameters
B) It is a non-parametric technique
C) It is a parametric technique
D) The response variable is usually assumed to be independently distributed and homoscedastic

Question 16

If you are interested in conducting a multiple hypotheses test to determine whether and are both unity for a regression , what would the restricted regression be?

A)
B)
C)
D)

Question 17

For question , you are given the following data
The regression equation is
Y_t = $\beta$ ₁ + $\beta$ ₂X_2t + $\beta$ ₃X_3t + u_t

-Which of the following is the correct value for ?

A) 2.89
B) 1.30
C) 0.84
D) We cannot determine the value of from the information given in the question

Question 18

What is the relationship, if any, between t-distributed and F-distributed random variables?

A) A t-variate with z degrees of freedom is also an F(1, z)
B) The square of a t-variate with z degrees of freedom is also an F(1, z)
C) A t-variate with z degrees of freedom is also an F(z, 1)
D) There is no relationship between the two distributions.

Question 19

Trying many variables in a regression without basing the selection of candidate variables on a financial or economic theory is popularly referred to as

A) Data fitting
B) Data clipping
C) Data mining
D) None of the above

Question 20

Which of these is a mathematical expression of the residual sum of squares?
(I) (II) (III)

A) I only
B) I and II only
C) I and III only
D) I, II and III